Michael Robinson Acknowledgements Collaborators: Brett Jefgerson, - - PowerPoint PPT Presentation

Michael Robinson

Filtrations of covers, Sheaves, and Integration

Michael Robinson

Acknowledgements

• Collaborators:

– Brett Jefgerson, Clifg Joslyn, Brenda Praggastis, Emilie Purvine

(PNNL)

– Chris Capraro, Grant Clarke, Griffjn Kearney, Janelle Henrich, Kevin

Palmowski (SRC)

– J Smart, Dave Bridgeland (Georgetown)

• Students:

– Philip Coyle – Samara Fantie

• Recent funding: DARPA, ONR, AFRL

– Robby Green – Fangei Lan – Michael Rawson – Metin Toksoz-Exley – Jackson Williams

Michael Robinson

Key ideas

• Motivate fjltrations of partial covers as

generalizing consistency fjltrations of sheaf assignments

• Explore fjltrations of partial covers as interesting

mathematical objects in their own right

• Instill hope that fjltrations of partial covers may be

(incompletely) characterized Big caveat: Only fjnite spaces are under consideration!

Michael Robinson

Consistency fjltrations

Michael Robinson

Context

• Assemble stochastic models of data locally into a

global topological picture

– Persistent homology is sensitive to outliers – Statistical tools are less sensitive to outliers, but cannot

handle (much) global topological structure

– Sheaves can be built to mediate between these two

extremes... this is what I have tried to do for the past decade or so

• The output is the consistency fjltration of a sheaf

assignment

Michael Robinson

Topologizing a partial order

Open sets are unions

• f up-sets

Michael Robinson

Topologizing a partial order

Intersections

• f up-sets are also

up-sets

Michael Robinson

( ) ( ) A sheaf on a poset is...

A set assigned to each element, called a stalk, and …

ℝ ℝ2 ℝ2 ℝ3

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1)

ℝ ℝ2 ℝ2

(1 -1)

( )

0 1 1 0

( )

• 3 3
• 4 4

This is a sheaf of vector spaces on a partial order

ℝ3

( )

0 1 1 1 0 1

ℝ

(-2 1)

Stalks can be measure spaces! We can handle stochastic data

2 3 1 2 -2 3 -3 1 -1

Michael Robinson

( ) ( ) A sheaf on a poset is...

… restriction functions between stalks, following the

• rder relation…

ℝ ℝ2 ℝ2 ℝ3

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1)

ℝ ℝ2 ℝ2

(1 -1) 2 3 1

( )

0 1 1 0

( )

• 3 3
• 4 4

2 -2 3 -3 1 -1 This is a sheaf of vector spaces on a partial order

ℝ3

( )

0 1 1 1 0 1

ℝ

(-2 1)

(“Restriction” because it goes from bigger up-sets to smaller ones)

Michael Robinson

( ) ( ) A sheaf on a poset is...

ℝ ℝ2 ℝ2 ℝ3

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1)

ℝ ℝ2 ℝ2

(1 -1)

( )

0 1 1 0

( )

• 3 3
• 4 4

This is a sheaf of vector spaces on a partial order

ℝ3

( )

0 1 1 1 0 1

ℝ

(-2 1) (1 -1) =

( )

0 1 1 1 0 1 (1 0)

= (0 1)( )

1 0 1 0 1 1

( )

0 1 1 0

( )

• 3 3
• 4 4

( )

1 0 1 0 1 1 2 -2 3 -3 1 -1 =

… so that the diagram commutes!

2 3 1 2 3 1 2 -2 3 -3 1 -1 2 -2 3 -3 1 -1

Michael Robinson

( ) ( ) An assignment is...

… the selection of a value on some open sets

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1) (1 -1) 2 3 1

( )

0 1 1 0

( )

• 3 3
• 4 4

2 -2 3 -3 1 -1

( )

0 1 1 1 0 1 (-2 1)

(-1)

( )

2 3

( )

• 4
• 3

( )

• 3
• 4

( )

3 2

(-4) (-4)

2 3

• 2
• 3
• 1

The term serration is more common, but perhaps more opaque.

Michael Robinson

( ) ( ) A global section is...

… an assignment that is consistent with the restrictions

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1) (1 -1) 2 3 1

( )

0 1 1 0

( )

• 3 3
• 4 4

2 -2 3 -3 1 -1

( )

0 1 1 1 0 1 (-2 1)

(-1)

( )

2 3

( )

• 4
• 3

( )

• 3
• 4

( )

3 2

(-4) (-4)

2 3

• 2
• 3
• 1

Michael Robinson

( ) ( ) Some assignments aren’t consistent

… but they might be partially consistent

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1) (1 -1) 2 3 1

( )

0 1 1 0

( )

• 3 3
• 4 4

2 -2 3 -3 1 -1

( )

0 1 1 1 0 1 (-2 1)

(+1)

( )

2 3

( )

• 4
• 3

( )

• 3
• 4

( )

3 2

(-4) (-4)

2 3 1

• 2
• 3
• 1

Michael Robinson

( ) ( ) Consistency radius is...

… the maximum (or some other norm) distance between the value in a stalk and the values propagated along the restrictions

0 1 1 1 0 1 1 0 1 0 1 1 (1 0) (0 1) (1 -1) 2 3 1

( )

0 1 1 0

( )

• 3 3
• 4 4

2 -2 3 -3 1 -1

( )

0 1 1 1 0 1 (-2 1)

(+1)

( )

2 3

( )

• 4
• 3

( )

• 3
• 4

( )

3 2

(-4) (-4)

2 3 1

• 2
• 3
• 1

2 3 1

( )

0 1 1 1 0 1

( )

3 2

• = 2

( )

2 3 (1 -1)

• 1 = 2

(+1) -

2 3 1

• 2
• 3 = 2 14
• 1

MAX ≥ 2 14

Note: lots more restrictions to check!

Michael Robinson

Amateur radio foxhunting

Typical sensors:

• Bearing to Fox
• Fox signal strength
• GPS location

Michael Robinson

Bearing sensors

. 5 1 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3

Antenna pattern

Michael Robinson

Bearing sensors… reality…

. 5 1 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3

Antenna pattern

Michael Robinson

Bearing observations

Bearing as a function of sensor position

. 5 1 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3

Antenna pattern Recenter

Michael Robinson

Bearing sheaf

Typical Mbearing function:

. 5 1 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3

Antenna pattern

Michael Robinson

Bearing sheaf (two sensors)

Fox position Sensor 2 position, Bearing Sensor 1 position, Bearing Fox position, Sensor 1 position Fox position, Sensor 2 position ℝ2×ℝ2 ℝ2×S1 ℝ2×S1 ℝ2×ℝ2 ℝ2 pr1 pr1 (pr2,Mbearing) (pr2,Mbearing) Global sections of this sheaf correspond to two bearings whose sight lines intersect at the fox transmitter

Michael Robinson

Consistency of proposed fox locations

Consistency radius minimization … … converges to a likely fox location … does not converge!

Michael Robinson

Local consistency radius

1 ½ ⅓

(2) (1) (1) (1) (½) Consistency radius of this open set = 0 Lemma: Consistency radius on an open set U is computed by

• nly considering open sets V1 ⊆ V2 ⊆ U

{A,C} {B,C} {C} {A,B,C}

Michael Robinson

Local consistency radius

1 ½ ⅓

(2) (1) (1) (1) (½) c(U) = ½ Lemma: Consistency radius does not decrease as its support grows: if U ⊆ V then c(U) ≤ c(V). Consistency radius of this open set = 0

Michael Robinson

Local consistency radius

1 ½ ⅓

(2) (1) (1) (1) (½) c(U) = ½ c(V) = ½ c(U ∩ V) = 0 Lemma: Consistency radius does not decrease as its support grows: if U ⊆ V then c(U) ≤ c(V).

Michael Robinson

Consistency radius is not a measure

1 ½ ⅓

(2) (1) (1) (1) (½) c(U) = ½ c(V) = ½ c(U ∩ V) = 0 c(U ∪ V) = ⅔ ≠ c(U) + c(V) – c(U ∩ V) (Consistency radius yields an inner measure after some work)

Michael Robinson

The consistency fjltration

1 ⅓

(1) (1)

Consistency threshold ½ ⅔

1 ½ ⅓

(2) (1) (1) (1) (½)

1 ½ ⅓

(2) (1) (1) (1) (½)

1 ½ ⅓

(2) (1) (1) (1) (½)

Consistency radius = ⅔

• … assigns the set of open sets (open cover) with consistency

less than a given threshold

• Lemma: consistency fjltration is itself a sheaf of collections
• f open sets on (ℝ,≤). Restrictions in this sheaf are cover

coarsenings.

refjne refjne

Michael Robinson

Filtrations of partial covers

Michael Robinson

Covers of topological spaces

• Classic tool: Čech cohomology

– Coarse – Usually blind to the cover; only sees the underlying space

• Cover measures (Purvine, Pogel, Joslyn, 2017)

– How fjne is a cover? – How overlappy is a cover?

Michael Robinson

Cover measures

• Theorem: (Birkhofg) The set of covers ordered by

refjnement has an explicit rank function

– The rank of a given cover is the number of sets in its

downset as an antichain of the Boolean lattice

– This counts the number of sets of consistent faces there are

• Conclusion: An assignment whose maximal cover has

a higher rank is more self-consistent

Michael Robinson

Cover measures

• Consider the following two covers of {1,2,3,4}

1 2 3 4

A

1 2 3 4

B

1 2 3 4 1 2 3 4 1 2 3 4

Refjne Refjne Refjne

Total = 6 sets Total = 11 sets Since 6 < 11, cover B is coarser

Michael Robinson

The lattice of covers

• Theorem: The lattice of

covers is graded using this rank function

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coarser Finer

Lattice graphic by E. Purvine

Michael Robinson

Defjning CTop : partial covers

• Start with a fjxed topological space
• Objects: Collections of open sets
• No requirement of coverage

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Michael Robinson

Defjning CTop : partial covers

• Morphisms are refjnements of

covers: If 𝒱 and 𝒲 are partial covers, 𝒲 refjnes 𝒱 if for all V in 𝒲 there is a U in 𝒱, with V ⊆ U.

• Convention: 𝒱 → 𝒲

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

Michael Robinson

Irredundancy

• Irredundant cover has no cover

elements contained in others

• Minimal representatives of

CTop isomorphism classes

– According to inclusion, not

refjnement

• Lemma: Every fjnite partial

cover is CTop-isomorphic to a unique irredundant one

1 2 3 4 5 6

Refjne

1 2 3 4 5 6

Michael Robinson

Defjning SCTop : Filtrations in CTop

• Objects are chains of morphisms in CTop with a

monotonic height function

– Height increases as cover coarsens – Could be the cover lattice rank,

but need not be

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

Michael Robinson

Defjning SCTop : Filtrations in CTop

• Morphisms are commutative ladders of refjnements

with a monotonic mapping ϕ : ℝ→ℝ of height functions

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.2 0.7 0.3 0.1 height 1 2 3 4 5 6

Refjne

Michael Robinson

Defjning SCTop : Filtrations in CTop

• Morphisms are commutative ladders of refjnements

with a monotonic mapping ϕ : ℝ→ℝ of height functions

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.2 0.7 0.3 0.1 height 1 2 3 4 5 6

Refjne

Michael Robinson

Defjning SCTop : Filtrations in CTop

• Morphisms are commutative ladders of refjnements

with a monotonic mapping ϕ : ℝ→ℝ of height functions

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.2 0.7 0.3 0.1 height 1 2 3 4 5 6

Refjne Refjne Refjne Refjne

Michael Robinson

Interleavings in SCTop

• Pair of morphisms between two objects

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.2 0.7 0.3 0.1 height 1 2 3 4 5 6

Refjne Refjne Refjne Refjne

Michael Robinson

Interleavings in SCTop

• Measure the maximum displacement of the heights,

minimize over all interleavings = interleaving distance

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.0 0.5 0.0 height

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Refjne Refjne

1.2 0.7 0.3 0.1 height 1 2 3 4 5 6

Refjne Refjne Refjne Refjne

Dist = 0.2 Dist = 0.3 Dist = 0.2 Dist = 0.2 Dist = 0.2 Dist = 0.3 Dist = 0.2 Dist = 0.1

Michael Robinson

Consistency fjltration stability

1 ⅓

(1) (1)

½ ⅔

1 ½ ⅓

(2) (1) (1) (1) (½)

1 ½ ⅓

(2) (1) (1) (1) (½)

1 ½ ⅓

(2) (1) (1) (1) (½)

Consistency radius = ⅔

• Theorem: Consistency fjltration is continuous under an

appropriate interleaving distance

• Thus the persistent Čech cohomology of the consistency

fjltration is robust to perturbations

Consistency threshold

refjne refjne

Michael Robinson

A small perturbation …

• Perturbations allowed in both assignment and

sheaf (subject to it staying a sheaf!) 1 ½ ⅓

(2) (1) (1) (1) (½)

0.9 0.6 0.3

(2) (1) (0.8) (0.8) (0.4) Max difgerence = 0.2

Michael Robinson

A small perturbation …

• Compute consistency fjltrations...

1 ½ ⅓

(2) (1) (1) (1) (½)

½ ⅔ Consistency threshold {C} {A,C} {B,C} {A,B,C}

0.9 0.6 0.3

(2) (1) (0.8) (0.8) (0.4)

0.6 0.66 Consistency threshold {C} {A,C} {B,C} {A,B,C} 0.3

Max difgerence = 0.2

Michael Robinson

… bounds interleaving distance

½ ⅔ Consistency threshold {C} {A,C} {B,C} {A,B,C} 0.6 0.66 Consistency threshold {C} {A,C} {B,C} {A,B,C} 0.3 {C} {A,C} {B,C} {A,B,C}

Shift = 0.5-0.3 = 0.2 Refjne Note that {A,C} ⊆ {A,B,C} etc.

Michael Robinson

… bounds interleaving distance

½ ⅔ Consistency threshold {C} {A,C} {B,C} {A,B,C} 0.6 0.66 Consistency threshold {C} {A,C} {B,C} {A,B,C} 0.3 {C} {A,C} {B,C} {A,B,C}

Shift = 0.6-0.5 = 0.1 Max shift = 0.2, This is bounded above by constant times the perturbation (0.2 in this case) Refjne

Michael Robinson

Summarizing a fjltration

Michael Robinson

Defjning Con : consistency functions

• Objects: order preserving functions Open(X) → ℝ+
• Example: local consistency radius

1 ½ ⅓

(2) (1) (1) (1) (½) {A,C} {B,C} {C} {A,B,C} ½ ½ ⅔ Open sets Assignment Local consistency radius Object in Con

Michael Robinson

Defjning Con : consistency functions

Ideally, we want…

• Consistency radius is a functor ShvFPA → Con
• A functor Con → SCTop acting by thresholding

{A,C} {B,C} {C} {A,B,C} ½ ½ ⅔ Open sets Local consistency radius Object in Con A C B A C B A C B

Refjne Refjne

Consistency fjltration Object in SCTop Height = consistency

Michael Robinson

Defjning Con : consistency functions

Ideally, we want…

• Consistency radius is a functor ShvFPA → Con
• A functor Con → SCTop acting by thresholding

To get this, the morphisms of Con are a little strange A morphism K: m → n of Con is a nonnegative real K so that m(U) ≤ K n(U) for all open U. Composition works by multiplication!

Michael Robinson

Defjning Con : consistency functions

A morphism K: m → n of Con is a nonnegative real K so that m(U) ≤ K n(U) for all open U.

½ ½ ⅔ Object in Con ½ ⅔ Object in Con 1 1 ½ 2 ½ These objects are not Con-isomorphic!

Michael Robinson

Con and SCTop

Theorem: Con is equivalent to a subcategory of SCTop by way of two functors:

• A faithful functor Con → SCTop
• A non-faithful functor SCTop → Con

such that Con → SCTop → Con is the identity functor. Interpretation: May be able to summarize fjltrations

• f partial covers using consistency functions, but this

is lossy!

Michael Robinson

Con → SCTop

• Motivation: generalization of consistency fjltration
• Idea: thresholding!

⅔ Object in Con 1 A C B A C B

Refjne Refjne

Object in SCTop A C B A C B

Refjne

½ ½ ⅔ 1 Showing an irredundant representative for this object {A,C} {B,C} {C} {A,B,C} Open sets

Michael Robinson

Con → SCTop

• Morphisms in Con transform to linear rescalings
• f the heights in SCTop … monotonicity does the

rest

½ ⅔ Morphism in Con 1 A C B A C B

Refjne Refjne

Morphism in SCTop A C B A C B

Refjne

½ ⅔ 1 ½ ½ ⅔ 2 A C B A C B A C B

Refjne Refjne

½ ⅔ 1

Michael Robinson

Con → SCTop

• Morphisms in Con transform to linear rescalings
• f the heights in SCTop … monotonicity does the

rest

½ ⅔ Morphism in Con 1 A C B A C B

Refjne Refjne

Morphism in SCTop A C B A C B

Refjne

½ ⅔ 1 ½ ½ ⅔ 2 A C B A C B A C B

Refjne Refjne

½ ⅔ 1

Michael Robinson

Con → SCTop

• Morphisms in Con transform to linear rescalings
• f the heights in SCTop … monotonicity does the

rest: Faithful!

½ ⅔ Morphism in Con 1 A C B A C B

Refjne Refjne

Morphism in SCTop A C B A C B

Refjne

½ ⅔ 1 ½ ½ ⅔ 2 A C B A C B A C B

Refjne Refjne

½ ⅔ 1

Michael Robinson

SCTop → Con

• At fjrst, this seems easy. Just look up the threshold

for each open set

A C B A C B

Refjne Refjne

Object in SCTop A C B A C B

Refjne

½ ⅔ 1 ⅔ Object in Con 1 ? {A,C} {B,C} {C} {A,B,C} Open sets

Michael Robinson

SCTop → Con

• At fjrst, this seems easy. Just look up the threshold

for each open set

A C B A C B

Refjne Refjne

Object in SCTop A C B A C B

Refjne

½ ⅔ 1 ⅔ Object in Con 1 ½ {A,C} {B,C} {C} {A,B,C} Open sets

Michael Robinson

SCTop → Con

• But what if the cover is not irredundant?
• This does not matter!

⅔ Object in Con 1 A C B

Refjne Refjne

Object in SCTop

Refjne

½ ½ ⅔ 1 A C B A C B A C B Fix: take the smallest threshold where the open set is contained in a cover element

Michael Robinson

SCTop → Con

½ A C B A C B

Refjne Refjne

Morphism in SCTop A C B A C B

Refjne

⅔ 1 A C B A C B A C B

Refjne Refjne

½ ⅔ 1 ⅔ Morphism in Con 1 ½ ½ ⅔

• Recall: SCTop morphisms are given by height

rescaling functions ϕ, which may not be linear

Michael Robinson

SCTop → Con

• Morphism in Con is given by K = max
• Not faithful!

½ A C B A C B

Refjne Refjne

Morphism in SCTop A C B A C B

Refjne

⅔ 1 A C B A C B A C B

Refjne Refjne

½ ⅔ 1 ⅔ Morphism in Con 1 ½ ½ ⅔ t ϕ(t) 2

Michael Robinson

Hope for a characterization

Michael Robinson

Pivoting to inner measures

• SCTop and Con are fairly elaborate

– Objects are “two-level”: open subsets, labeled with real

values

• It’s potentially useful to study their interaction with

functions on the underlying space

– The way to do this is via integration – But then we need to transform Con objects into something

like a measure… they have no hope of being additive!

• One way to do that is through an endofunctor that

creates inner measures, Inner : Con→Con

Michael Robinson

Inner measures from Con

Theorem: Suppose that m is an object in Con, a consistency function. A Borel inner measure is generated by Im(V) = m(U). An inner measure satisfjes

• I(∅) = 0
• I(U) ≥ 0
• I(U ∪ V) ≥ I(U) + I(V) – I(U ∩ V)

U ⊆ V

Σ

Note: this is an endofunctor since if m(U) ≤ K n(U), then Im(U) ≤ K In(U) by linearity

Michael Robinson

Integration w.r.t. inner measures

• This works like Baryshnikov-Ghrist real-valued Euler

integration (strong connection to sheaves!)

• For an f : X → ℝ and inner measure I, defjne

f dI = lim [I(f-1([z/n,∞))) – I(f-1((-∞,-z/n]))]

• Theorem: (Monotonicity) 0 ≤ f ≤ g implies f dI ≤ g dI
• Theorem: (Partial linearity) (r f) dI = r f dI

and if 0 ≤ f ≤ g, (f + g) dI ≥ f dI + g dI

Σ

1 n z = 1 ∞

n→∞

∫

Sublevel sets Superlevel sets Resolution

∫ ∫ ∫ ∫ ∫ ∫ ∫

Michael Robinson

Missing structure

Cannot remove the inequality, though. There are functions f, g for which (f + g) dI ≠ f dI + g dI Additionally,

• (f g) dI is not an inner product
• | f |p dI is not a norm, and
• | f – g |p dI 1/p is not a pseudometric.

Is there a topology induced by an inner measure?

∫ ∫ ∫ ∫ ∫ ∫

Michael Robinson

Open questions...

• What’s the interaction between topological

invariants (Euler characteristic/integral), geometric invariants inner measures, and fjltrations?

– Expect a sheaf-theoretic answer!

• How robust are inner measures to distortions or

stochastic variability?

– Use interleaving distance on SCTop! – How does that push forward to inner measures?

• What does the integral of a function on a base

space of a sheaf assignment mean?

Michael Robinson

More open questions...

• Can we relate structure of local consistency of a

sheaf assignment to the structure of functions on the base?

• Do all inner measures arise from consistency

functions?

– They defjnitely don’t have to come from sheaf

assignments!

• What is the most natural topology on the space of

inner measures?

• Is there a better notion of measures for consistency

functions? Hopefully an actual measure?

Michael Robinson

To learn more...

Michael Robinson michaelr@american.edu http://drmichaelrobinson.net Preprint: https://arxiv.org/abs/1805.08927 Software: https://github.com/kb1dds/pysheaf

Michael Robinson

Excursion! Friday evening ~ 6pm

• RetroComputing Society of Rhode Island